Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
Ab, Ac = the orthogonal projections of A' on AC. AB. resp.
Na, N1 = the NPC centers of A'AbAc, AAbAc, resp.
Similarly Nb, N2 and Nc, N3.
A*B*C* = the triangle bounded by the perpendicular bisectors of NaN1, NbN2, NcN3.
Which is the locus of P such that:
1. 1. NaN1, NbN2, NcN3 are concurrent?
H lies on the locus.
1.2. NaN1, NbN2, NcN3 bound a triangle A#B#C# parallelogic with ABC?
O, I lie on the locus.
2. NaNbNc, N1N2N3 are orthologic?
O, H lie on the locus.
3. ABC, A*B*C* are perspective?
H lies on the locus.
4. A'B'C', A*B*C* are perspective?
H lies on the locus.
5. ABC, A*B*C* are orthologic?
H, O, I lie on the locus.
6. A'B'C', A*B*C* are orthologic ?
H, O, I lie on the locus.
> 1. 1. NaN1, NbN2, NcN3 are concurrent at Z11?
Quartic Q* with trilinear equation:
Q* = a*((SB^2-SC^2)*SA*S^2*b*c*u^4+ (S^2+SA^2)*((2*S^4-(5*SB+3*SC) *SA*S^2+2*(SB+SC)*SA^2*SC)*v^2 -(2*S^4-(3*SB+5*SC)*SA*S^2+2*( SB+SC)*SA^2*SB)*w^2)*v*w+S^2*S A*v*w*(SB-SC)*((8*R^2*SA+S^2)* u^2-(8*R^2+SA)*b*c*v*w)) + … = 0
through ETC’s: 4, 20, 523
Z11(H) = X(12241)
Z11( X(20) ) = {5,6} /\ {25,69}
= 2*a^8-(b^2+c^2)*a^6-(b^2-c^2)^ 2*a^4+(b^2+c^2)^3*a^2-(b^4-c^4 )^2 : : (barycentrics)
= 5*X(3618)-3*X(11245)
= On lines: {3,5596}, {5,6}, {25,69}, {30,3313}, {66,1368}, {140,5157}, {141,206}, {193,6997}, {311,460}, {343,1974}, {511,6756}, {524,9969}, {599,10154}, {1176,7499}, {1351,7528}, {1503,5907}, {3589,11548}, {3618,7539}, {3620,7493}, {3818,3867}, {5159,6697}, {5480,13142}, {5921,6816}, {5972,6698}, {6248,6748}, {6776,7395}, {7529,11898}, {7819,10547}, {9715,10519}, {9967,12134}
= midpoint of X(9967) and X(12134)
= reflection of X(13142) in X(5480)
= {X(141), X(206)}-Harmonic conjugate of X(6676)
= [ 0.420865358812950, -0.86032842275108, 4.042030916667611 ]
> 1.2. NaN1, NbN2, NcN3 bound a triangle A#B#C# parallelogic with ABC?
O, I lie on the locus.
Locus = Q* \/ {Darboux cubic K004}
Parallelogic centers:
For P=I
Z12(A->A#) = X(7)
Z12(A#->A) = (b+c)*a^7-(3*b^2+2*b*c+3*c^2)* a^6+(b+c)*(2*b^2-b*c+2*c^2)*a^ 5+(2*b^4+2*c^4+b*c*(b+c)^2)*a^ 4-3*(b^2-c^2)^2*(b+c)*a^3+(b^2 +c^2)*(b^2+4*b*c+c^2)*(b-c)^2* a^2+(b^2-c^2)*(b-c)^3*b*c*a-(b ^2-c^2)^2*(b-c)^2*b*c : : (barycentrics)
= [ 1.629407573383575, 1.76289640574726, 1.668163474828474 ]
For P=O
Z12(A->A#) = X(2)
Z12(A#->A)= X(6688)
> 2 NaNbNc, N1N2N3 are orthologic? O, H lie on the locus.
Locus = {Euler line } \/ {circle with center O2 and radius=|R^2-2*SW|*R/|5*R^2-2*S W| (no ETCs on it)}
O2 = {2,3} /\ {195,511}
= (a^8-2*(b^2+c^2)*a^6-3*b^2*c^ 2*a^4+(b^2+c^2)*(2*b^4+b^2*c^2 +2*c^4)*a^2-(b^4+c^4)*(b^2-c^2 )^2)*a : : (trilinears)
= X(54)-3*X(6030) = 3*X(381)-4*X(13160) = 5*X(1656)-4*X(5576) = 7*X(3526)-8*X(7568) = 7*X(3851)-6*X(7565)
= Shinagawa coefficients: (5*E+8*F, -9*E-8*F)
= On lines: {2,3}, {49,10625}, {54,6030}, {110,10627}, {156,2979}, {195,511}, {399,2918}, {524,13432}, {542,3519}, {568,10984}, {1147,13340}, {1204,8717}, {1216,10540}, {1385,9591}, {1495,5447}, {1503,9920}, {1614,6101}, {2883,9919}, {2889,9143}, {2917,6000}, {3053,9700}, {3098,10539}, {3311,9683}, {3579,9626}, {5012,10263}, {5446,13353}, {5462,13339}, {5663,7691}, {6455,9682}, {6800,9704}, {7755,11063}, {8193,12645}, {10117,10282}, {10575,10620}, {11255,12220}, {12310,12359}
= midpoint of X(7691) and X(8718)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,22,2937), (3,2937,2070), (3,3830,7503), (3,3843,7514), (3,5073,7526), (3,5899,5), (3,7387,381), (3,7517,1656), (3,9909,7506), (3,12083,382), (4,7492,7525), (4,7525,3), (5,12088,5899), (20,7502,3), (22,10323,26), (26,10323,3), (550,7555,7488), (1656,7517,7545), (6636,12088,5), (7509,7530,3851), (7526,12082,5073)
= [ 131.398484588813300, 130.18233214580200, -147.131019890792300 ]
Orthologic centers:
For P=O:
Z2(Na->N1) = complement of X(10610)
= ((b^2+c^2)*a^2-(b^2-c^2)^2)*(a ^6-3*(b^4+b^2*c^2+c^4)*a^2+2*( b^4-c^4)*(b^2-c^2)) : : (baricentrics)
= 3*X(2)+X(6288) = 9*X(2)-X(12254) = 3*X(5)-X(3574) = X(54)-5*X(1656) = X(195)-9*X(5055) = 3*X(381)+X(7691) = 3*X(547)-X(8254) = 3*X(1209)+X(3574) = 3*X(6288)+X(12254) = 3*X(10610)-X(12254)
= On lines: {2,6288}, {3,7703}, {5,51}, {54,1656}, {140,13470}, {195,5055}, {252,12060}, {381,7691}, {403,11017}, {498,12956}, {499,12946}, {539,547}, {858,11592}, {1216,11808}, {1493,2888}, {1594,11576}, {2072,12363}, {2917,7514}, {3519,5056}, {3628,5972}, {3851,12307}, {5071,12325}, {5449,13363}, {5576,13391}, {5663,13160}, {5790,7979}, {5886,12785}, {5907,11802}, {5943,10115}, {5965,12812}, {6145,9833}, {6152,7577}, {6153,10170}, {7393,9920}, {7579,7999}, {7730,11444}, {7741,13079}, {9777,12316}, {9977,11178}, {10255,12606}, {11230,12266}
= midpoint of X(i) and X(j) for these {i,j}: {5,1209}, {1216,11808}, {1493,2888}, {5907,11802}, {6288,10610}, {12606,13368}
= reflection of X(i) in X(j) for these (i,j): (973,13365), (6689,3628)
= complement of X(10610)
= {X(2), X(6288)}-Harmonic conjugate of X(10610)
= [ 3.795402228301171, -1.53296166402141, 2.950221528552506 ]
Z2(N1->Na) = {4,12006} /\ {548,3589}
= (13*cos(2*A)-9/2)*cos(B-C)-4*c os(A)*cos(2*(B-C))-10*cos(A)-2 *cos(3*A) : : (trilinears)
= On lines: {4,12006}, {548,3589}
= [ 1.716891503639485, 1.20742501050912, 2.012343396029074 ]
> 3. ABC, A*B*C* are perspective at Z3? H lies on the locus.
Locus: {Quartic through X(30) } \/ { circum-quintic through H}
Z3(H) = {2,6} /\ {5, 389}
= (b^2+c^2)*a^4-2*(b^2-c^2)^2*a^ 2+(b^4-c^4)*(b^2-c^2) : : (barycentrics)
= 3*X(2)+X(6515)
= On lines:
{2,6}, {3,12241}, {4,64}, {5,389}, {11,11436}, {15,465}, {16,466}, {19,5928}, {20,1192}, {24,161}, {25,1503}, {30,11438}, {32,441}, {34,10361}, {51,125}, {52,11585}, {53,2052}, {54,10018}, {68,6642}, {92,1146}, {140,578}, {143,13371}, {154,6353}, {182,6676}, {184,468}, {185,235}, {186,12022}, {189,7365}, {226,6708}, {275,6749}, {287,1915}, {297,3981}, {306,3965}, {324,338}, {329,6554}, {397,470}, {398,471}, {403,5890}, {406,5706}, {428,11550}, {429,5799}, {440,573}, {458,7745}, {461,3332}, {472,5321}, {473,5318}, {511,1368}, {541,1539}, {549,11430}, {550,13403}, {568,2072}, {569,7542}, {572,7536}, {576,5159}, {580,7515}, {631,11425}, {800,6509}, {858,3060}, {860,5721}, {1147,13292}, {1151,1589}, {1152,1590}, {1181,3542}, {1196,6388}, {1204,1885}, {1209,7405}, {1350,7386}, {1352,5020}, {1353,5972}, {1498,3089}, {1583,11090}, {1584,11091}, {1585,3070}, {1586,3071}, {1587,3535}, {1588,3536}, {1591,12239}, {1592,12240}, {1593,6696}, {1594,3567}, {1595,10110}, {1596,6000}, {1620,3522}, {1656,11432}, {1848,2262}, {1861,1864}, {1906,11381}, {1990,11547}, {1995,11442}, {3066,6997}, {3098,10691}, {3168,6530}, {3517,9833}, {3526,11426}, {3541,10982}, {3564,6677}, {3796,7493}, {3832,11469}, {3867,9969}, {3925,11435}, {4232,11206}, {5012,13394}, {5067,11431}, {5085,7494}, {5094,9777}, {5097,6723}, {5133,5640}, {5432,11429}, {5713,7532}, {5810,7535}, {5816,7522}, {6389,6617}, {6619,10002}, {6823,9729}, {7392,10516}, {7505,7592}, {7506,12134}, {7583,8968}, {7715,13419}, {8254,12234}, {8263,8681}, {8991,11473}, {9820,12161}, {10257,13352}, {10272,12227}, {10594,11457}, {11402,12007}, {13142,13346}
= midpoint of X(i) and X(j) for these {i,j}: {4,10605}, {25,1899}, {125,12828}, {394,6515}
= reflection of X(9306) in X(6677)
= complementary conjugate of X(6389)
= isotomic conjugate of X(801)
= polar conjugate of X(1105)
= complement of X(394)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2,343,141), (2,1993,11064), (2,3580,343), (2,6515,394), (2,10601,3589), (2,11433,6), (5,389,12233), (51,125,427), (51,427,5480), (184,468,10192), (184,11245,8550), (185,235,2883), (468,11245,184), (1146,6354,92), (1204,1885,5894), (5449,5462,5), (6353,6776,154), (8550,10192,184)
= [ 2.003672347070931, 1.65214493599261, 1.572099981418296 ]
>4. A'B'C', A*B*C* are perspective? H lies on the locus.
Locus is complicated. H does not lie on the locus
>5. ABC, A*B*C* are orthologic? H, O, I lie on the locus.
Locus is complicated.
For P=H
Z5(A->A*)=H
Z5(A*->A)={4,64} /\ {6,20}
= 2*a^10-(b^2+c^2)*a^8-8*(b^4-b^ 2*c^2+c^4)*a^6+10*(b^4-c^4)*(b ^2-c^2)*a^4-2*(b^4-c^4)^2*a^2- (b^4-c^4)*(b^2-c^2)^3 : : (barycentrics)
= 3*X(51)-X(1885) = 3*X(389)-X(13403) = 3*X(428)-X(11381) = 3*X(12241)-2*X(13403)
= On lines: {2,1192}, {3,12233}, {4,64}, {5,4550}, {6,20}, {25,2883}, {30,143}, {51,1885}, {54,10295}, {141,6815}, {185,1503}, {235,5893}, {376,11425}, {427,1204}, {428,11381}, {516,12432}, {524,5889}, {548,11430}, {550,578}, {973,974}, {1350,10996}, {1498,7487}, {1593,5480}, {1595,3357}, {1598,5878}, {1620,3523}, {1657,11432}, {1890,12688}, {1899,12173}, {2777,10110},…
= midpoint of X(185) and X(3575)
= reflection of X(4) in X(11745)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (4,10605,6247), (427,1204,6696), (5480,5894,1593), (7689,7706,5)
= [ 7.379957619051008, 7.38054772827993, -4.875079769540653 ]
For P=I
Z5(A->A*)=X(7)
Z5(A*->A)= ((b+c)*a^6-(b+c)*(2*b^2-b*c+2* c^2)*a^4-4*b*c*(b^2+3*b*c+c^2) *a^3+(b^2-c^2)*(b-c)*(b^2+4*b* c+c^2)*a^2+(b^2-c^2)^2*(b+c)*b *c)/(-a+b+c) : : (barycentrics)
= On lines: {57,4008}
= [ 0.280546584500594, 0.47413196839003, 3.182936234021775 ]
For P=O
Z5(A->A*)=X(2)
Z5(A*->A)= a^2*((b^2+c^2)*a^6-3*(b^4+4*b^ 2*c^2+c^4)*a^4+(b^2-3*c^2)*(3* b^2-c^2)*(b^2+c^2)*a^2-(b^4-16 *b^2*c^2+c^4)*(b^2-c^2)^2) : : (barycentrics)
= 2*X(4)+X(9729) = 4*X(5)-X(13348) = X(51)+3*X(3839) = X(143)+5*X(546) = 2*X(143)-5*X(10110) = 3*X(373)+X(3543) = 5*X(381)-X(5891) = X(389)+5*X(3843) = 2*X(546)+X(10110) = 3*X(546)+X(13451)
= On lines: {3,10219}, {4,5943}, {5,13348}, {30,6688}, {51,3839}, {143,546}, {373,3543}, {381,511}, {389,3843}, {3060,5907}, {5480,8681},…
= midpoint of X(i) and X(j) for these {i,j}: {4,5943}, {3060,5907}
= reflection of X(3) in X(10219)
= [ -2.150811111664215, -2.86639404943082, 6.617773183050735 ]
> 6. A'B'C', A*B*C* are orthologic ? H, O, I lie on the locus.
Locus is complicated.
For P=H
Z6(A’->A*)=X(4)
Z6(A*->A’)=X(12241)
For P=O
Z6(A’->A*)=X(2)
Z6(A*->A’)= a^2*((b^2+c^2)*a^6-3*(b^4+4*b^ 2*c^2+c^4)*a^4+(b^2-3*c^2)*(3* b^2-c^2)*(b^2+c^2)*a^2-(b^4-16 *b^2*c^2+c^4)*(b^2-c^2)^2) : : (trilinears)
= 2*X(4)+X(9729) = 4*X(5)-X(13348) = X(51)+3*X(3839) = X(143)+5*X(546) = 2*X(143)-5*X(10110) = 3*X(373)+X(3543) = 5*X(381)-X(5891) = X(389)+5*X(3843) = 2*X(546)+X(10110) = 3*X(546)+X(13451)
= midpoint of X(4) and X(5943)
= reflection of X(3) in X(10219)
= On lines: {3,10219}, {4,5943}, {5,13348}, {30,6688}, {51,3839}, {143,546}, {373,3543}, {381,511}, {389,3843},…
= [ -2.150811111664215, -2.86639404943082, 6.617773183050735 ]
For P=I
Z6(A’->A*)=X(7)
Z6(A*->A’)= ((b+c)*a^5-(b^2+6*b*c+c^2)*a^4 +3*b*c*(b+c)*a^3+b*c*(b+3*c)*( 3*b+c)*a^2-(b^2-3*b*c+c^2)*(b+ c)^3*a+(b^2-c^2)*(b-c)*(b^3+c^ 3))/(-a+b+c)*a : : (barycentrics)
= On line {2809,12573}
= [ 0.654241106718235, 0.82823202127791, 2.765315648691257 ]
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